Functions

Functions:

    A function is a fundamental concept in mathematics, often described as a "black box" that processes an input to produce a corresponding output. You don't necessarily need to understand the internal workings of a function to use it—simply provide an input, and the function will return the appropriate output.

Example:
    f(x) = y
In this example, x is the input (independent variable), and y is the output (dependent variable) produced by the function f.

Consider the function:
    f(x) = 2x + 5

To evaluate the function for x = 5:
    f(5) = 2 × 5 + 5
            = 10 + 5
            = 15

Therefore, f(5) = 15.

Statistics

Statistics:

    Statistics is a branch of mathematics that deals with the collection, analysis, interpretation, presentation, and organization of data. It provides methods for making decisions and predictions based on data. The subject is broadly divided into two main areas:

1. Descriptive Statistics – Focuses on summarizing and organizing data using measures such as mean, median, mode, standard deviation, and graphical representations like histograms and box plots.

2. Inferential Statistics – Involves making predictions or inferences about a population based on a sample of data. It includes hypothesis testing, confidence intervals, regression analysis, and probability distributions.

Statistics is widely applied in fields like economics, medicine, engineering, social sciences, and business for data-driven decision-making. It helps in identifying patterns, testing hypotheses, and estimating probabilities.

Relation

 Relation:

                A relation R over a set X can be seen as a set of ordered pairs (x, y) of members of X.

               X= {1,2,3,4,5,6,7}

               the relation "is less than" on the set X is given by R= {(1,2), (2,3), (3,4), (4,5), (5,6), (6,7)}

                If we consider social relationships like fatherhood, brotherhood, sonhood, etc. 

                For example, Ram is the son of Dashrath, Sita was the daughter of Janak, etc.

 

Types of Relations:

• Empty Relation: 

                    A relation R on a set A is called Empty if the set A is empty set.

• Reflexive Relation:

                 A relation R on a set A is called reflexive if (a, a) € R holds for every element a € A i.e., if set A = {a, b} then R = {(a, a), (b, b)} is reflexive relation.

• Irreflexive relation:

                A relation R on a set A is called irreflexive if no (a, a) € R holds for every element a € A i.e., if set A = {a, b} then R = {(a, b), (b, a)} is irreflexive relation. 

 

• Symmetric Relation:

                A relation R on a set A is called symmetric if (b,a) € R holds when (a,b) € R.i.e. The relation R= {(4,5), (5,4), (6,5), (5,6)} on set A= {4,5,6} is symmetric.

 

• Transitive Relation:

                A relation R on a set A is called transitive if (a,b) € R and (b,c) € R then (a,c) € R for all a,b,c € A.i.e. Relation R={(1,2),(2,3),(1,3)} on set A={1,2,3} is transitive.

 

• Asymmetric relation:

                Asymmetric relation is opposite of symmetric relation. A relation R on a set A is called asymmetric if no (b,a) € R when (a,b) € R.

Set

 Set:

         A "set" is an undefined term,but we define it as a collection of well defined objects.

        e.g. A=collection of all the students in class 5 .

P= Collection of keys in a keychain.

Etc...

Types of sets:

Null set:

      A set having no elements is known as null set.

Singleton set:

      The set having only one element is known as singleton set.

Representation of set:

   There are two ways to represent a set 

    1-Tabular form

    2-Roster form

 1-Tabular Form:

      In this form of representation sets are basically represented by capital letters (A,B,C....Z), the elements are represented by small letters (a,b,c...z) and the elements are present within a curli braces "{}" and separated by commas","  .

      A={a,b,c}  ,  P={1,2,3}  etc.

2-Roster Form:

    There are some sets which are difficult to represent in tabular form as the number of elements are vast. like the set of real numbers,sets of rational etc.

    These sets are easily representable by the help of   Roster form .

  e.g.   R={x: x is a rela number}

           N={x: x is a natural number}

Subset:

  If there are 2 sets and all the elements of the 1st set are in the 2nd one the we say that the 1st set is the subset of the 2nd one .

            A={a,b,c,d}

            B={a,b,c,d,e,f}

  Then we say that ,   A ⊆ B

And we use the symbol "⊂" to represent the proper subset.

Mathematically,

    If A ⊆ B,

    Then  x ∈ A => x ∈ B

Power set:

    The set of all the subsets of a set is called as power set of that particular set.

 e.g.

A={a,b,c}

then power set of A 

P(A)={ { },{a},{b},{c},{a,b],{a,c},{b,c},{a,b,c} }

if the number of elements in a set is "n" then the number of elements on it`s power set is 2^n.

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